The Nuts and Bolts of Proofs

The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs

4th Edition - November 25, 2011

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  • Author: Antonella Cupillari
  • Paperback ISBN: 9780123822178
  • eBook ISBN: 9780123822185

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The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive and proofing by contradiction. It also includes the negation statement, and/or. It examines various theorems, such as the if and only-if, or equivalence theorems, the existence theorems, and the uniqueness theorems. In addition, use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are covered in this chapter. The book also provides mathematical topics for practicing proof techniques. Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book invaluable.

Key Features

  • Jumps right in with the needed vocabulary—gets students thinking like mathematicians from the beginning
  • Offers a large variety of examples and problems with solutions for students to work through on their own
  • Includes a collection of exercises without solutions to help instructors prepare assignments
  • Contains an extensive list of basic mathematical definitions and concepts needed in abstract mathematics


A supplement for undergraduate courses in higher mathematics as a primer to assist engineering and physical science students on fundamental proof techniques

Table of Contents

  • Chapter 1. Getting Started

    Introduction and Basic Terminology

    General Suggestions

    Chapter 2. Basic Techniques to Prove If/Then Statements

    What Does “If/Then” Mean?

    The Negation of a Statement: AND/OR

    Proof by Contrapositive

    Proof by Contradiction

    Chapter 3. Special Kinds of Theorems

    “If and Only If” or “Equivalence Theorems”

    Use of Counter Examples

    Mathematical Induction

    Existence Theorems

    Uniqueness Theorems

    Composite Statements

    Equality of Numbers

    Chapter 4. Some Mathematical Topics on Which to Practice Proof Techniques

    Basic Set Theory and Indexed Families

    About Functions


    The Basics of Groups


    Sizes of Infinity

    Chapter 5. Review Exercises

    Exercises without Solutions

    Collection of “Proofs”

    Solutions for the Exercises at the End of the Sections and the Review Exercises

    Solutions for the Exercises at the End of the Sections

    Solutions for the Review Exercises

    Other Books on the Subject of Proofs and Mathematical Writing

Product details

  • No. of pages: 296
  • Language: English
  • Copyright: © Academic Press 2012
  • Published: November 25, 2011
  • Imprint: Academic Press
  • Paperback ISBN: 9780123822178
  • eBook ISBN: 9780123822185

About the Author

Antonella Cupillari

Antonella Cupillari is an associate professor of mathematics at Pennsylvania State Erie in Behrend College. She received her Laurea in Mathematics in Italy, and her M.A. and Ph.D. at the State University of New York at Albany. She has been a participant in the Mathematical Association of America/National Science Foundation Institute on the "History of Mathematics and Its Use in Teaching." Cupillari is the author of several papers in analysis, mathematics education, and the history of mathematics. She is also the author of the first edition of The Nuts and Bolts of Proofs.

Affiliations and Expertise

Associate Professor of Mathematics, Pennsylvania State Erie in Behrend College, Erie, PA, USA

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